MA111: Contemporary mathematics . Jack Schmidt University of - - PowerPoint PPT Presentation

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MA111: Contemporary mathematics

Jack Schmidt

University of Kentucky

October 19, 2011

Schedule: HW Ch 5 Part Two is due Today, Oct 19th. HW Ch 5 Part Three is due Mon, Oct 24th. Exam 3 is Monday, Oct 24th, during class. Exams not graded yet (this week is busy; but will be done for midterms) Look for practice exam tonight; work it for Friday Today we will fix graphs without Euler paths or circuits

5.5: The theorems

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Theorem (Euler, 1736)

. . The sum of the degrees in a graph is twice the number of edges, so is even. .

Theorem (Euler, 1736 and Hierholzer, 1873)

. . A graph has an Euler circuit if and only if it is connected and every vertex has even degree. .

Corollary

. . A graph has an Euler path if and only if it is connected and exactly two of its vertices have odd degree. But what if we really want an Euler circuit??

5.7: Just do it

Exhaustive route - a circuit that uses all the edges at least once Optimal exhaustive route - an exhaustive route of shortest possible length Security guard and postman have to cover all the roads

“But boss, there are vertices of odd degree!” “I’ll give you a vertex of odd degree if you don’t get out there!”

If we can’t get a perfect answer (Euler circuit), then we get a “best” answer (Optimal Exhaustive Route) Eulerizing: Double edges on the graph until no odd degrees

5.7: How to do it?

The book wimps out: “just try it” The real answer was a revolutionary invention from 1965:

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Apply the Floyd-Warshall algorithm to find shortest paths . .

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Pair up bad vertices so that the total distance is minimized Edmonds, 1965

Our problems will be small enough that “just try it” should work However, there is a nice algorithm to handle grids: just walk along the edge of the grid, and double if you need to